- 19,943
- 11,029
Definition/Summary
A second order polynomial equation in one variable, its general form is [itex]ax^2 + bx + c = 0,[/itex] where x is the variable and a, b, and c are constants, and [itex]a \ne 0.[/itex]
Equations
[tex]ax^2 + bx + c = 0[/tex]
Extended explanation
Since a quadratic equation is a second degree polynomial equation, then the fundamental theorem of algebra states that two complex roots exist, counting multiplicity.
There are various analytical methods used for finding the roots of quadratic equations, one of the most common methods is the so-called quadratic formula and is derived by completing the square on the general expression shown above. The quadratic formula may be written thus,
[tex]x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\hspace{2cm}a\neq0[/tex]
The term under the square root is known as the discriminant and can be used to determine the form of the roots of the quadratic equation.
If [itex]b^2-4ac > 0[/itex] then there are two distinct real roots. Furthermore if the discriminant is a perfect square, then the two roots are also rational.
If [itex]b^2-4ac = 0[/itex] then there is one repeated real root.
If [itex]b^2-4ac < 0[/itex] then there are two distinct non-real roots. These two roots are the complex conjugate of each other.
* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
A second order polynomial equation in one variable, its general form is [itex]ax^2 + bx + c = 0,[/itex] where x is the variable and a, b, and c are constants, and [itex]a \ne 0.[/itex]
Equations
[tex]ax^2 + bx + c = 0[/tex]
Extended explanation
Since a quadratic equation is a second degree polynomial equation, then the fundamental theorem of algebra states that two complex roots exist, counting multiplicity.
There are various analytical methods used for finding the roots of quadratic equations, one of the most common methods is the so-called quadratic formula and is derived by completing the square on the general expression shown above. The quadratic formula may be written thus,
[tex]x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\hspace{2cm}a\neq0[/tex]
The term under the square root is known as the discriminant and can be used to determine the form of the roots of the quadratic equation.
If [itex]b^2-4ac > 0[/itex] then there are two distinct real roots. Furthermore if the discriminant is a perfect square, then the two roots are also rational.
If [itex]b^2-4ac = 0[/itex] then there is one repeated real root.
If [itex]b^2-4ac < 0[/itex] then there are two distinct non-real roots. These two roots are the complex conjugate of each other.
* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!